@MISC{Yu910somestochastic, author = {Yaming Yu}, title = {Some stochastic inequalities for weighted sums}, year = {910} }

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Abstract

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let Yi be i.i.d. random variables on R+. Assuming that log Yi has a log-concave density, we show that ∑ aiYi is stochastically smaller than ∑ biYi, if (log a1,...,log an) is majorized by (log b1,..., log bn). On the other hand, assuming that Y p i has a log-concave density for some p> 1, we show that ∑ aiYi is stochastically larger than ∑ biYi, if (a q 1,..., aqn) is majorized by (bq1,...,bq n), where p−1 + q−1 = 1. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko (1998) on Weibull variables is proved. Some applications in reliability and wireless communications are mentioned.